A regular hexagon and an equilateral triangle have equal perimeters. what is the ratio of the area of the hexagon to the area of the triangle? express your answer as a common fraction.

In this item, we let the perimeter of both polygons be P. The lengths of each side are calculated below.

Hexagon: s = (p/6) = p/6

Triangle: s = (p/3) = p/3

The areas of each polygon are also calculated below. It is noted that the polygons are regular (meaning, each side and angle are equal).

Area of Hexagon: A = 3√3/2 a²

Substituting the known values,

A = 3√3/2 (P/6) ²

Simplifying,

A = √3/24P²

For the triangle,

A = √3/4a²

Substituting,

A = √3/4 (P/3) ²

Simplifying,

A = √3/36P²

The ratio is equal to:

ratio = (√3/24P²) / (√3/36P²)

ratio = 3/2

ratio = 1.5